How do I prove the following using Fermat's theorem on sums of two squares?

Question

For naturals $x_i,\,y_i$ with $1\le i\le 2019$, $\prod_i (2x_i^2+3y_i^2)$ is not a perfect square.

Answer 1

you might as well assume that each pair $(x_i, y_i)$ is coprime, otherwise you just divide out by a long string of squares. Then your product is not divisible by any prime $r \equiv 13, 17,19,23 \pmod {24}.$ There is no restriction on prime factors $q \equiv 1,7 \pmod {24}.$

However, if we let $M$ be the set of primes represented by $2 x^2 + 3 y^2,$ we see that $M$ consists of $2,3$ themselves, then all $p \equiv 5, 11 \pmod{24}.$ The rule for numbers represented as $2u^2 + 3 v^2$ is that the sum of the exponents of primes from $M$ must be odd. That is for a single number. You have an odd count of those, so it is still true that the sum of exponents of primes from $M$ in the prime factorization of your big number must be odd. In particular, there is then at least one such prime factor with odd exponent, and your big product is not a square.

The basic theory here is treated, for example, in Cox, Primes of the Form $x^2 + n y^2$

Side note. A number $x^2 + 6 y^2$ with $\gcd(x,y) = 1$ has the sum of $M$ exponents even.

Primes $2 u^2 + 3 v^2$

  2,      3,      5,     11,     29,     53,     59,     83,    101,    107,
131,    149,    173,    179,    197,    227,    251,    269,    293,    317,
347,    389,    419,    443,    461,    467,    491,    509,    557,    563,
587,    653,    659,    677,    683,    701,    773,    797,    821,    827,
941,    947,    971,   1013,   1019,   1061,   1091,   1109,   1163,   1181,

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numbers $2 x^2 + 3 y^2$ with $\gcd(x,y) = 1$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primitive_go
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2 
2 0 3
Discriminant  24
Maximum number represented? 
1000
                 2 = 2
                 3 = 3
                 5 = 5
                11 = 11
                14 = 2 * 7
                21 = 3 * 7
                29 = 29
                30 = 2 * 3 * 5
                35 = 5 * 7
                50 = 2 * 5^2
                53 = 53
                59 = 59
                62 = 2 * 31
                66 = 2 * 3 * 11
                75 = 3 * 5^2
                77 = 7 * 11
                83 = 83
                93 = 3 * 31
                98 = 2 * 7^2
               101 = 101
               107 = 107
               110 = 2 * 5 * 11
               125 = 5^3
               131 = 131
               146 = 2 * 73
               147 = 3 * 7^2
               149 = 149
               155 = 5 * 31
               158 = 2 * 79
               165 = 3 * 5 * 11
               173 = 173
               174 = 2 * 3 * 29
               179 = 179
               194 = 2 * 97
               197 = 197
               203 = 7 * 29
               206 = 2 * 103
               210 = 2 * 3 * 5 * 7
               219 = 3 * 73
               227 = 227
               237 = 3 * 79
               242 = 2 * 11^2
               245 = 5 * 7^2
               251 = 251
               254 = 2 * 127
               269 = 269
               275 = 5^2 * 11
               290 = 2 * 5 * 29
               291 = 3 * 97
               293 = 293
               302 = 2 * 151
               309 = 3 * 103
               317 = 317
               318 = 2 * 3 * 53
               341 = 11 * 31
               347 = 347
               350 = 2 * 5^2 * 7
               354 = 2 * 3 * 59
               363 = 3 * 11^2
               365 = 5 * 73
               371 = 7 * 53
               381 = 3 * 127
               386 = 2 * 193
               389 = 389
               395 = 5 * 79
               398 = 2 * 199
               413 = 7 * 59
               419 = 419
               434 = 2 * 7 * 31
               435 = 3 * 5 * 29
               443 = 443
               446 = 2 * 223
               453 = 3 * 151
               461 = 461
               462 = 2 * 3 * 7 * 11
               467 = 467
               482 = 2 * 241
               485 = 5 * 97
               491 = 491
               498 = 2 * 3 * 83
               509 = 509
               515 = 5 * 103
               525 = 3 * 5^2 * 7
               530 = 2 * 5 * 53
               539 = 7^2 * 11
               542 = 2 * 271
               557 = 557
               563 = 563
               579 = 3 * 193
               581 = 7 * 83
               587 = 587
               590 = 2 * 5 * 59
               597 = 3 * 199
               605 = 5 * 11^2
               606 = 2 * 3 * 101
               626 = 2 * 313
               635 = 5 * 127
               638 = 2 * 11 * 29
               642 = 2 * 3 * 107
               651 = 3 * 7 * 31
               653 = 653
               659 = 659
               669 = 3 * 223
               674 = 2 * 337
               677 = 677
               683 = 683
               686 = 2 * 7^3
               701 = 701
               707 = 7 * 101
               723 = 3 * 241
               725 = 5^2 * 29
               734 = 2 * 367
               749 = 7 * 107
               750 = 2 * 3 * 5^3
               755 = 5 * 151
               770 = 2 * 5 * 7 * 11
               773 = 773
               786 = 2 * 3 * 131
               795 = 3 * 5 * 53
               797 = 797
               803 = 11 * 73
               813 = 3 * 271
               818 = 2 * 409
               821 = 821
               827 = 827
               830 = 2 * 5 * 83
               866 = 2 * 433
               869 = 11 * 79
               875 = 5^3 * 7
               878 = 2 * 439
               885 = 3 * 5 * 59
               894 = 2 * 3 * 149
               899 = 29 * 31
               914 = 2 * 457
               917 = 7 * 131
               926 = 2 * 463
               930 = 2 * 3 * 5 * 31
               939 = 3 * 313
               941 = 941
               947 = 947
               957 = 3 * 11 * 29
               965 = 5 * 193
               971 = 971
               974 = 2 * 487
               995 = 5 * 199

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